Another important property of this manifold is that it is orientable, which loosely (in Euclidean space) means that moving a coordinate basis along the manifold does not change its orientation (which could mean, for instance, the determinant changing sign). A classic example of a non-orientable surface in Euclidean space is the Möbius strip: a normal vector moving along this surface will return to its original location, but face in the opposite direction. This explanation of orientable is intentionally loose, since defining it precisely would require an understanding of local frames for the tangent bundle, which is a tool that is not needed for this paper. However, orientable is a quality needed for the method of integration on \(\mathcal {M}\)  outlined in this paper, so it must be established. This is immediate from Proposition 15.5 in [1]}, and the assumption that \(\nabla _\theta g \ne 0\) .
